Školjka

%V0 Let $\mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $f: \mathbb{N} \mapsto \mathbb{N}$ satisfying $$f(m + f(n)) = n + f(m + 95)$$ for all $m$ and $n$ in $\mathbb{N}.$ What is the value of $\sum^{19}_{k = 1} f(k)?$

%V0 Let $a > 2$ be given, and starting $a_0 = 1, a_1 = a$ define recursively: $$a_{n+1} = \left(\frac{a^2_n}{a^2_{n-1}} - 2 \right) \cdot a_n.$$ Show that for all integers $k > 0,$ we have: $\sum^k_{i = 0} \frac{1}{a_i} < \frac12 \cdot (2 + a - \sqrt{a^2-4}).$

%V0 Let $a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that (a) The polynomial $p(x) = x^{n} - a_{1}x^{n - 1} + ... - a_{n - 1}x - a_{n}$ has preceisely 1 positive real root $R$. (b) let $A = \sum_{i = 1}^n a_{i}$ and $B = \sum_{i = 1}^n ia_{i}$. Show that $A^{A} \leq R^{B}$.

%V0 Find all surjective functions $f: \mathbb{N} \mapsto \mathbb{N}$ such that for every $m,n \in \mathbb{N}$ and every prime $p,$ the number $f(m + n)$ is divisible by $p$ if and only if $f(m) + f(n)$ is divisible by $p.$ Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran

%V0 For every $n\in\mathbb{N}$ let $d(n)$ denote the number of (positive) divisors of $n$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ with the following properties: $d\left(f(x)\right) = x$ for all $x\in\mathbb{N}$. $f(xy)$ divides $(x - 1)y^{xy - 1}f(x)$ for all $x$, $y\in\mathbb{N}$. Proposed by Bruno Le Floch, France

%V0 Let $P\!\left(x\right)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n\!\left(x\right) = x$ is equal to $P\!\left(n\right)$ for every $n \geqslant 1$, where $T^n$ denotes the $n$-fold application of $T$. Proposed by Jozsef Pelikan, Hungary